Упр.5.491 ГДЗ Виленкин Жохов 5 класс Часть 2, Просвещение (Математика)

Задача

а) 61/64 — (7/12 — 5/14) · (13/16 + 1/2); в) 1 — 1/2 + 1/3 — 1/4 + 1/5 — 1/6;
б) (1 — 11/17) · (3/4 — 5/12 + 11/18); г) 1/8 + 3/8 + 1/12 + 5/12 + 1/16 + 7/16 + 1/20 + 9/20.

Подробный ответ

а) Преобразуем скобки:

$$ \frac{7}{12}-\frac{5}{14}=\frac{49}{84}-\frac{30}{84}=\frac{19}{84}, \qquad \frac{13}{16}+\frac{1}{2}=\frac{13}{16}+\frac{8}{16}=\frac{21}{16}. $$

Тогда

$$ \frac{61}{64}-\left(\frac{7}{12}-\frac{5}{14}\right)\cdot\left(\frac{13}{16}+\frac{1}{2}\right) = \frac{61}{64}-\frac{19}{84}\cdot\frac{21}{16}. $$

$$ \frac{19}{84}\cdot\frac{21}{16}=\frac{19\cdot 1}{4\cdot 16}=\frac{19}{64}, $$

$$ \frac{61}{64}-\frac{19}{64}=\frac{42}{64}=\frac{21}{32}. $$

б)

$$ \left(1-\frac{11}{17}\right)\cdot\left(\frac{3}{4}-\frac{5}{12}+\frac{11}{18}\right) = \frac{6}{17}\cdot\left(\frac{27}{36}-\frac{15}{36}+\frac{22}{36}\right). $$

$$ \frac{27}{36}-\frac{15}{36}+\frac{22}{36}=\frac{34}{36}. $$

Тогда

$$ \frac{6}{17}\cdot\frac{34}{36}=\frac{2}{3}. $$

в)

$$ 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6} = \frac{60}{60}-\frac{30}{60}+\frac{20}{60}-\frac{15}{60}+\frac{12}{60}-\frac{10}{60}. $$

$$ =\frac{30}{60}+\frac{20}{60}-\frac{15}{60}+\frac{12}{60}-\frac{10}{60} =\frac{50}{60}-\frac{15}{60}+\frac{12}{60}-\frac{10}{60} $$

$$ =\frac{35}{60}+\frac{12}{60}-\frac{10}{60} =\frac{47}{60}-\frac{10}{60} =\frac{37}{60}. $$

г)

$$ \frac{1}{8}+\frac{3}{8}+\frac{1}{12}+\frac{5}{12}+\frac{1}{16}+\frac{7}{16}+\frac{1}{20}+\frac{9}{20} $$

Сгруппируем слагаемые:

$$ \left(\frac{1}{8}+\frac{3}{8}\right)+\left(\frac{1}{12}+\frac{5}{12}\right)+\left(\frac{1}{16}+\frac{7}{16}\right)+\left(\frac{1}{20}+\frac{9}{20}\right). $$

$$ =\frac{4}{8}+\frac{6}{12}+\frac{8}{16}+\frac{10}{20} =\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}=2. $$